“Risk” for financial institutions generally means uncertainty of asset values in the future. It is very important for financial institutions to appropriately manage risks in various financial transactions. The largest risk for, in particular, banks among financial institutions is credit risk. Banks need to independently establish strong credit risk management systems in order to maintain their financial soundness. Supervisory authorities are required to verify the appropriateness of a bank's credit risk management systems on the basis of the second pillar of the “Basel II”, which is an international agreement concerning bank supervisory methods. In view of such external and internal demands, it can be said that risk management departments should play an important role in bank management to appropriately manage credit risks. In order to appropriately carry out credit risk management, it is necessary to quantitatively grasp information such as “how much loss occurs at how high a probability over the next one year from a certain portfolio”.
In order to quantitatively grasp such information, it is effective to perform risk management taking into account the probabilistic nature. This can be achieved by appropriately grasping, for each loss after a fixed period assumed based on the portfolio, the probability of occurrence of the loss. FIG. 3 is an explanatory diagram showing a loss expected in a fixed period in future as a graph. In FIG. 3, probabilities of occurrence of respective losses that are assumed after a fixed period are shown with respect to the respective losses. Information necessary for the credit risk management described above can be grasped by quantitatively predicting such a probability distribution. A mathematical procedure for reasonably predicting, from various data, a probability density function or a distribution function for giving such a probability distribution is called “measurement (or estimation) of risks” in the field of financial engineering.
When a probability density function, or cumulative distribution function, of losses is accurately predicted (i.e., a risk is “measured”), various risk measures are calculated as indicators for portfolio management practice. For example, “expected loss (EL)” is calculated, “credit VaR (Value at Risk)” is calculated, and “unexpected loss (UL)” is calculated. The expected loss (EL) represents a loss expected as an average after a certain period. The credit VaR is a single measure represented by an amount and represents a loss where the distribution function takes a fixed value. This fixed value is called a confidence coefficient or a confidence interval. For example, a value such as 99% or 99.97% is selected as the fixed value. Actually, the credit VaR can be considered as the upper limit of a loss that should be assumed in operation management after operation in that period. In the present specification, the credit VaR may be simply referred to as VaR. The unexpected loss (UL) is calculated by a difference between the credit VaR and the expected loss. When these specific measures are calculated, a risk management department can quantitatively grasp beforehand, for example, whether a provision equivalent to an expected loss can be prepared or whether equity capital can be appropriated for a loss equivalent to an unexpected loss exceeding the provision. A portfolio operation department can reasonably determine, by using credit risk measures of its own company as reference information, whether the portfolio should be held or the portfolio should be rebalanced.
In predicting the probability distribution of losses, i.e., measuring risks, mainly two ideas are known depending on the premise of how a loss is recognized, i.e., what kind of event is considered to be losses. One idea is to grasp only a fall in credit value caused by default. This is called the default mode. Another idea is to grasp a fall in credit value due to a fall in creditworthiness of a borrower company in addition to default. This is called the MtM (Mark to market) mode.
FIG. 4 is a table in which characteristics of the default mode and the MtM mode are summarized. The default mode is an idea on an accrual basis in which only the occurrence of default is regarded as a loss. The default mode is used by traders who mainly aim at maturity transactions in evaluating, for example, a portfolio mainly including lendings and guarantees. On the other hand, the MtM mode is an idea on a market price basis in which the losses caused by a fall in market prices of individual credits are taken into account. The MtM mode is used by traders who mainly aim at short-term trades in evaluating, for example, a portfolio mainly including bonds and derivatives.
A credit value is recognized on a book value basis in the default mode and is recognized on a market price basis in the MtM mode. The event in which loss is recognized as credit risk is default in the default mode and includes a fall in credit rating in the market in addition to default in the MtM mode.
Because of such a difference, selection of the main parameters used for the mathematical procedure used in measurement of risk is also different between the default mode and the MtM mode. In the default mode, the exposure of respective companies, the probability of default (PD), the correlation in changes of creditworthiness, and the like are taken into account as the parameters. The exposure of a company means a loss that occurs when the company is in default and can be calculated from a transaction balance and a loss-at-default rate. The correlation in changes of creditworthiness is a parameter representing association of credit capabilities of a plurality of companies and is calculated from, for example, time series data of stock prices and bond prices. On the other hand, in the MtM mode, information on the cash flow of a transaction, market information, a rating transition probability, the correlation in changes of creditworthiness, and the like are take into account. The cash flow information of a transaction means information on a cash flow generated by interest payment (coupon) of a bond or the like forming a portfolio. The market information means the information on the market that affects market prices such as yield curves given for each rating of bonds. Since such a market price in the market depends on credit rating, in risk management in the MtM mode, the rating transition probability, which is the probability of transition from a credit rating at the present point to respective credit ratings after a fixed period, is needed.
An example of the ideas of loss in which the default mode explained above is adopted as a model is Credit Risk+ (Credit Suisse Financial Products). An example in which the MtM mode is adopted as a model is Credit Metrics (JP Morgan).
Measurement of credit risk will now be explained. The measurement of credit risk is performed in two steps. First, there is a step of describing the probabilities of loss of respective companies in the measurement of credit risk. Then, there is a step of adding up losses of the respective companies contained in a portfolio. FIGS. 5 and 6 are diagrams showing concepts of loss of a company. FIG. 5 shows a concept in the case of the default mode and FIG. 6 shows a concept in the case of the MtM mode. As described above, in the default mode, it is assumed that loss occurs only when a default occurs. In the MtM mode, it is assumed that loss occurs because of a fall in credit rating after a risk measurement period in addition to default. In the MtM mode, loss is different depending on the credit rating after the risk measurement period. Profit could occur instead of loss.
FIG. 7 is an explanatory diagram showing a concept of adding up losses of companies. The concept only of the default mode is explained below. It is assumed that the probability of default for each loss is known for each of plural companies (company “i” and company “j”). In FIG. 7, the probability that the company “i” falls into default is 5% and the loss at that time is 100. The probability that the company “j” falls into default is also 5% and the loss at that time is 50. Even in such a case, in order to add up the losses of the plurality of companies and clarify the probability distribution of the losses, it is necessary to further take into account the correlation regarding default events between the companies. This is for the purpose of accurately reflecting the correlation of business performance or the like between the companies included in a real portfolio. As shown in FIG. 7, when there is no correlation between the companies (in the case of correlation=0), default events independently occur. Therefore, the probability that both the companies fall in default and the total loss reaches 150 is represented by the product of the default probability of the company “i” and the default probability of the company “j” as 0.25%. On the other hand, when there is a complete correlation between the companies (in the case of correlation=1), the default of one company leads to the default of the other company. Therefore, the probability that the total loss amount reaches 150 is the same as the default probability 5% of one company. Besides these cases, in FIG. 7, the cases in which there is a partial correlation between the companies, i.e., correlation coefficient is 0.2 and 0.5 is described. In this way, it is possible to calculate, concerning the plurality of companies, the probabilities of losses appropriately taking into account correlation.
FIG. 8 shows such probabilities for each loss for one company (the company “i”) and in a case in which the correlation in the case of two companies (the company “i” and the company “j”) is 0.2. The graphs shown in FIG. 8 indicate the probability density function by which the probabilities of respective losses are obtained. The probability density function corresponds to the probability density function in FIG. 3. In principle, it is evident that the probability distribution of losses of a portfolio with an arbitrary number of companies can be calculated. Once the probability density function is calculated, the distribution function shown in FIG. 9 can be calculated by integrating the probability density function. For example, the amount where the distribution function indicates the value 99% is the amount of credit VaR. Therefore, the measures such as EL, UL, and credit VaR can be calculated by calculating a probability distribution. Actually, unlike the example shown in FIG. 8, in actual operation of a portfolio and management of banks, it is not easy to take into account all of several tens of thousands to several million credits included in banks.
A credit risk measurement model will now be explained. As described above, in order to grasp the credit risk of a portfolio, it is necessary to appropriately take into account the correlation between companies (obligors) observed in default and the change of creditworthiness. Therefore, it is necessary to mathematically model the credit risk such that the correlation can be taken into account. In the mathematical modeling of the credit risk, it is necessary to model the credit risk such that the correlation observed in default and the change in creditworthiness of the companies can be appropriately reflected. As a credit risk measurement model employing such a correlation between companies, in recent years, a model called “firm-value model” is often used.
Among the firm-value model, there are one-factor firm-value model and multi-factor firm-value model. The one-factor firm-value model represents a firm value after a fixed period using the following equation:
[Formula 1]Zi=αi X+√{square root over (1−α12)}εi  (1)where, i is the suffix indicating a company (an individual company) and Zi indicates the firm value of the company “i”. X on the right side is called common risk factors and αi is called sensitivity coefficient. The common risk factors X are factors common to companies for which a risk is measured by the credit risk measurement model. As an assumption of the firm-value model, the common risk factors X are random variables conforming to the standard normal distribution N(0, 1). εi is a factor representing fluctuation in each of the companies, which cannot be explained by the factor X common to the companies, and is called idiosyncratic factor. It is assumed that the idiosyncratic factor εi also conforms to the standard normal distribution N(0, 1). The coefficient in front of the idiosyncratic factor is set so that the firm value Zi conforms to the standard normal distribution N(0, 1).
As an important assumption in the firm-value model, X and εi are random variables independent from each other. That is,
[Formula 2]X˜N(0, 1), εi˜N(0, 1) Corr(X, εi)=0, Corr(εi, εj)=0 (i≠j)  (2)From these random variables, it is possible to calculate the correlation between the firm values as follows:[Formula 3]Corr(Zi,Zj)=αiαj (i≠j)  (3)This indicates that, in the one-factor firm-value model, the correlation between firm values of different companies is generated through the presence of the common risk factors X that affects the firm values of all the companies.
The multi-factor firm-value model is obtained by modeling a credit risk assuming that a plurality of common risk factors exists. The firm value Zi of the company “i” is represented as follows:
[Formula 4]
                              Z          i                =                                            ∑                              j                =                1                                            N                F                                      ⁢                                          α                ij                            ⁢                              X                j                                              +                                                    1                -                                                      ∑                                          j                      =                      1                                                              N                      F                                                        ⁢                                      α                    ij                    2                                                                        ⁢                          ɛ              i                                                          (        4        )            where, NF is the number of the common risk factors. It is assumed that, in the firm-value model, different common risk factors Xk and X1 are independent from each other. That is, the following equation holds.[Formula 5]Corr(Xk,X1)=δk1 Corr(Xk,εi)=0  (5)δk1 is Kronecker Delta. The αi in Equation 1 and αij in Equation 4 are parameters that include the information concerning the correlation between the companies. There is known, for example, a method of estimating the parameters from a principal component analysis of the time series data of industry share price indexes.
In the one-factor or multi-factor firm-value model based on such assumptions, it is assumed that the company “i” falls into default when the firm value Zi -falls below a fixed level. For example, in the case of the default mode, when the value of firm value causing default (default threshold) is represented as Ci, since the firm value Zi conforms to the standard normal distribution, the probability PDi that the company falls into default is as follows:
[Formula 6]
                                                                        PD                i                            =                              Pr                ⁢                                  {                                                            Z                      i                                        <                                          C                      i                                                        }                                                                                                        =                              Φ                ⁡                                  (                                      C                    i                                    )                                                                                        (        6        )            The firm value Zi depends on the common risk factors, which are random variables. The probability PDi is an average over the common risk factors. Therefore, the probability PDi is referred to as average default probability of the company “i”. Conversely, when the average default probability PDi is given, the default threshold Ci is determined as follows:[Formula 7]Ci=Φ−1(PDi)  (7)where, Φ−1 on the right side is the inverse function of the distribution function of the standard normal distribution. The default probability of the company “i” is estimated, for example, from the credit rating separately given for the company “i”. It is possible to set the default threshold by giving the average default probability in this way. FIG. 10 is an explanatory diagram showing a concept of the default threshold given in this way. When the firm value Z conforming to the standard normal distribution falls below the default threshold Ci after a fixed period (e.g., one year), a default occurs.
When the default threshold Ci is determined from the average default probability, the conditional default probability can be calculated under the condition that the values of the common risk factors are determined. For example, the conditional default probability of the company “i” is given by the following equation:
[Formula 8]
                                          PD            i                    ⁡                      (                          x              ->                        )                          =                              Pr            ⁢                          {                                                                                          Z                      i                                        <                                          C                      i                                                        ❘                                      X                    ->                                                  =                                  x                  ->                                            }                                =                      Φ            (                                                            C                  i                                -                                                      α                    ->                                    ·                                      x                    ->                                                                                                1                  -                                                                                                          α                        ->                                                                                    2                                                                        )                                              (        8        )            where, x is a realization of the common risk factors X and represents the condition for giving the conditional default probability. In the one-factor firm-value model, x is a scalar amount. In the multi-factor firm-value model, x is a vector amount including respective common risk factors as elements. In the one-factor and multi-factor firm-value models, the idiosyncratic factors (ε) are independent from each other for each of the companies. Therefore, under the condition that common risk factors take certain realizations x, defaults of the respective companies occur independently from each other. This characteristic is called conditional independence and is a characteristic extremely beneficial in numerical calculation.
In the MtM mode, the market price fluctuation due to credit rating transition is taken into account in recognizing profit and loss in addition to default in the default mode. Therefore, thresholds are set on the basis of rating transition probabilities to divide the firm value Zi into hierarchies. That is,
[Formula 9]pi(r→s)Ci(r→s)  (9)The upper formula in formula (9) represents the transition probability of the company “i” from a credit rating r to a credit rating s. The lower formula represents the threshold for separating the credit rating s and a credit rating s−1. FIG. 11 is an explanatory diagram for explaining a method by which the credit rating and the default of a company are determined according to the firm value after one year. By considering the thresholds and the transition probabilities shown in the figure, the formulation applied to the default mode can be easily extended to the MtM mode.
A specific method for credit risk measurement after the modeling will now be explained. To calculate the probability distribution of losses, in principle, the losses and occurrence probabilities only have to be calculated for all scenarios considering credit states (scenarios) assumed for borrower companies after a fixed period. However, when it is attempted to execute this calculation, in reality, an extremely difficult calculation is necessary. For example, when the portfolio to be calculated is a combination of credits given to ten thousand borrowers, the number of combinations that should be considered as the states of the respective borrowers after one year is the 10000th power of 2 (250th power of one trillion). It can be said that it is practically impossible to calculate the losses and occurrence probabilities in all the scenarios. Therefore, to create a loss distribution, some approximation calculation is necessary. As a standard method for the approximation calculation, there is the Monte Carlo simulation method.
The Monte Carlo simulation method is a general method for performing an analysis of a probabilistic phenomenon by performing a numerical simulation using random numbers. To apply the Monte Carlo simulation method in the firm-value model, for example, steps 1 to 5 described below are sequentially executed. As step 1, first, the common risk factors are determined. This processing is a processing for generating random numbers conforming to the standard normal distribution and determining the values of the common risk factors. As step 2, the default probabilities of respective companies are calculated. This is a processing for calculating the conditional default probabilities of the respective companies under the value of the common risk factors determined in step 1. As step 3, default scenarios are created. This processing is a processing for creating the scenarios representing the credit states of the companies after one year. Specifically, the uniform random number in the interval [0, 1] is generated for each of the companies and, when the random number falls below the default probability calculated in step 2, it is determined that a default has occurred. As the following step 4, losses are calculated. The losses caused from the entire portfolio are calculated on the basis of the scenarios created in step 3. In the Monte Carlo simulation method, the step 5 of repeating steps 1 to 4 multiple times is executed to count, for each loss, the frequency of occurrence of events that cause the loss. Consequently, it is possible to create the loss distribution.
Besides the Monte Carlo simulation method, there is known a method of calculating the distribution of losses using the Laplace transform. A process for calculating a loss distribution using the Laplace transform will now be explained concerning an example of the default mode. The density function of a loss distribution is represented as fL(t). The Laplace transform of thereof is give by the following equation:
[Formula 10]
                                                        f              ^                        L                    ⁡                      (            λ            )                          =                                            ∫              0              ∞                        ⁢                                          ⅇ                                                      -                    λ                                    ⁢                                                                          ⁢                  t                                            ⁢                                                f                  L                                ⁡                                  (                  t                  )                                            ⁢                                                          ⁢                              ⅆ                t                                              =                      E            ⁡                          [                              exp                (                                                      -                    λ                                    ⁢                                                                          ⁢                                      L                    ~                                                  ⁢                                                                  )                            ]                                                          (        10        )            
In particular, the rightmost side of this equation is the function called moment generating function in the field of probability statistics. Respective expressions used here have the following meanings:
[Formula 11]
                                          L            ~                    ⁢                      :                    ⁢                                          ⁢                                                                                                                              ⁢                                      random                    ⁢                                                                                  ⁢                    variable                    ⁢                                                                                  ⁢                    representing                                    ⁢                                                                                                                                                              loss                  ⁢                                                                          ⁢                  of                  ⁢                                                                          ⁢                  the                  ⁢                                                                          ⁢                  entire                  ⁢                                                                          ⁢                  portfolio                                                              ⁢                                          ⁢                      (                                          L                ~                            =                                                ∑                                      i                    =                    1                                    N                                ⁢                                                      E                    i                                    ⁢                                                            D                      ~                                        i                                                                        )                          ⁢                                  ⁢                              E            i                    ⁢                      :                    ⁢                                          ⁢          loss          ⁢                                          ⁢          of          ⁢                                          ⁢          the          ⁢                                          ⁢          firm          ⁢                                          ⁢                      “            i            ”                    ⁢                                          ⁢          at          ⁢                                                            ⁢                                                          ⁢          default          ⁢                                          ⁢                      (            Exposure            )                          ⁢                                  ⁢                  N          ⁢                      :                    ⁢                                          ⁢          the          ⁢                                          ⁢          number          ⁢                                          ⁢          of          ⁢                                          ⁢          firms          ⁢                                          ⁢          in          ⁢                                          ⁢          the          ⁢                                          ⁢          portfolio                ⁢                                  ⁢                                            D              ~                        i                    ⁢                      :                    ⁢                                          ⁢                                                                      definition                  ⁢                                                                          ⁢                  function                                                                                                                          for                    ⁢                                                                                  ⁢                    the                    ⁢                                                                                  ⁢                    firm                    ⁢                                                                                  ⁢                                          “                      i                      ”                                                        ⁢                                                                                                                      ⁢                                          ⁢                      (                                                                                                      Default                      ⁢                                            ⁢                      1                                        ,                                                                                                                    Non                    ⁢                                          -                                        ⁢                    Default                    ⁢                                        ⁢                    0                                                                        )                                              (        11        )            
From this result, it is possible to calculate the distribution of losses by subjecting the moment generating function to Laplace inversion. When there is no correlation between the companies, the moment generating function is represented as follows:
[Formula 12]
                              E          ⁡                      [                          exp              ⁡                              (                                                      -                    λ                                    ⁢                                                                          ⁢                                      L                    ~                                                  )                                      ]                          =                              ∏                          i              =              1                        N                    ⁢                      (                          1              -                              p                i                            +                                                p                  i                                ⁢                                                                  ⁢                                  exp                  ⁡                                      (                                                                  -                        λ                                            ⁢                                                                                          ⁢                                              E                        i                                                              )                                                                        )                                              (        12        )            where, pi is the default probability of the company “i”, which is the same as PDi in formula (6).
On the other hand, when the correlation among the companies is taken into account by the firm-value model, the moment generating function is calculated as follows from the characteristic of conditional independence:
[Formula 13]
                                                        f              ^                        L                    ⁡                      (            λ            )                          =                              ∫                                          x                ->                            ∈                              R                                  N                  F                                                              ⁢                                    ∏                              i                =                1                            N                        ⁢                                          {                                  1                  -                                                            p                      i                                        ⁡                                          (                                              x                        ->                                            )                                                        +                                                                                    p                        i                                            ⁡                                              (                                                  x                          ->                                                )                                                              ⁢                                          exp                      ⁡                                              (                                                                              -                            λ                                                    ⁢                                                                                                          ⁢                                                      E                            i                                                                          )                                                                                            }                            ⁢                                                ϕ                                      N                    F                                                  ⁡                                  (                                      x                    ->                                    )                                            ⁢                              ⅆ                                  x                  ->                                                                                        (        13        )            where,[Formula 14]pi({right arrow over (x)}): Conditional default probability of the firm “i” under the condition that “a common risk factor is {right arrow over (X)}={right arrow over (x)}” (same as PDI({right arrow over (x)}) in formula (8))φNF({right arrow over (x)}): Density function of NF-dimensional standard normal distribution  (14)
Such calculation of a loss distribution using the Laplace transform method can be performed in two steps. First, as step 1, the moment generating function is calculated. As step 2, the Laplace inversion of the moment generating function is executed. The Laplace transform method has been explained for the default mode as an example. However, risk measurement in the MtM mode can also be executed by the Laplace transform method by changing the moment generating function. Even when a model other than the firm-value model is used, the probability density function and the distribution function that give the probability distribution of losses can be calculated by the Laplace transform method if the moment generating function can be calculated. Therefore, it is possible to measure the credit risk of a portfolio.
An example of the techniques for executing credit risk calculation with an analytical method using Fourier transform rather than the Laplace transform method is disclosed in Japanese Patent Application Laid-Open No. 2000-148721. Examples of theoretical researches for executing calculation of a credit risk using Laplace transform are disclosed in Martin, R., K. Thompson and C. Browne, “Taking to the saddle,” Risk, 14(6), 2001, pp. 91 to 94, and Glasserman, P. and J. Ruiz-Mata, “Computing the credit loss distribution in the Gaussian copula model: a comparison of methods, “Journal of credit risk, 2(4), 2006, pp. 33 to 66.